Hydrodynamics of GRB After-Glow

The detection of delayed emission at X-ray optical and radio wavelengths, ``after-glow'', suggests that the relativistic shell which emitted the initial GRB due to internal shocks decelerates on encountering an external medium, giving rise to the after-glow. At a very early stage (few seconds), the observed bolometric luminosity increases as t^2. On longer time scales (more than about 10s), the luminosity drops as t^{-1}. If the main burst is long enough, an intermediate stage of constant luminosity will form. In this case, the after-glow overlaps the main burst, otherwise there is a time separation between the two. On the long time scale, the flow decelerate in a self similar way, reaching non relativistic velocities after about 30days. Due to the deceleration and the accumulation of ISM material, the relation between the observed time, the shock radius, and its Lorentz factor, is given by t=R/16\gamma^2c which is a factor of 8 different from the usual expression. The majority of particles are those of the original ejecta (and not the ISM) up to about 900s. These particles reach sub-relativistic velocities on a time scale of about 2hours, well before the flow becomes sub-relativistic. Therefore the ejecta particles are probably unimportant for most of the after-glow radiation. We show that even though only a small fraction of the energy is given to the electrons, most of the energy can be radiated over time. If this fraction is greater than 10% radiation losses will significantly influence the hydrodynamical evolution at early times (less than 1day).